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Stochastic inflation in numerical relativity

Yoann L. Launay1,*, Gerasimos I. Rigopoulos2,†, and E. Paul S. Shellard1,‡

  • 1Centre for Theoretical Cosmology, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
  • 2School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom

  • *Contact author: yoann.launay@outlook.com
  • †Contact author: gerasimos.rigopoulos@ncl.ac.uk
  • ‡Contact author: eps1@cam.ac.uk
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Phys. Rev. D 113, 123515 – Published 9 June, 2026

DOI: https://doi.org/10.1103/h2cb-q1mt

Abstract

A set of 3+1 equations for stochastic inflation incorporating all metric and scalar matter degrees of freedom, first presented in previous work Launay et al. [Phys. Rev. D 109, 123523 (2024).], are rederived in a gauge invariant manner. We then present numerical implementations of these stochastic equations, cast in the Baumgarte-Shapiro-Shibata-Nakamura formulation of numerical relativity, demonstrating their efficacy in both a slow-roll and an ultra slow-roll scenario. We find the evolution is correctly reproduced for all the dynamical variables, and the energy and momentum constraints are well satisfied. This demonstrates that the stochastic equations are theoretically and numerically robust and ready to be applied to a wider inflationary landscape. Our simulations result in real space realizations of the fully nonlinear stochastic dynamics with gradients and anisotropic expansion retained. This work generalizes standard stochastic inflation, inflationary numerical relativity, and lattice cosmology, opening up the possibility for reliable predictions of nonperturbative phenomena and providing precise initial conditions for subsequent cosmological eras.

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Phys. Rev. D 113, 123515– Published 9 June, 2026
  • Received 26 January 2026
  • Accepted 22 April 2026
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